Heba was asked to find this integral using $u$-substitution:
$$
\int(-12 x-1) \sqrt{-6 x^{2}-x+1} d x
$$

How should Heba define $u$ ?

Choose 1 answer:
(A) $u=-6 x^{2}-x+1$
(B) $u=\sqrt{-6 x^{2}-x+1}$
(c) $u=-12 x-1$
(D) $u=\sqrt{x}$

Question

Heba was asked to find this integral using $u$-substitution:
$$
\int(-12 x-1) \sqrt{-6 x^{2}-x+1} d x
$$

How should Heba define $u$ ?

Choose 1 answer:
(A) $u=-6 x^{2}-x+1$
(B) $u=\sqrt{-6 x^{2}-x+1}$
(c) $u=-12 x-1$
(D) $u=\sqrt{x}$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the integral using $u$-substitution, we need to identify a function inside the integral that, when differentiated, resembles the rest of the integrand. ‖ ‖ step 2 ⋮ Looking at the integrand $(-12x-1)\sqrt{-6x^2-x+1}$, we notice that the derivative of $-6x^2-x+1$ is $-12x-1$, which is present outside the square root. ‖ ‖ step 3 ⋮ Therefore, we should choose $u$ to be the expression inside the square root, $u = -6x^2-x+1$, so that $du$ will be proportional to $-12x-1 \, dx$. ‖ ‖ step 4 ⋮ This choice will simplify the integral, as the substitution will cancel out the $-12x-1$ term, leaving us with an integral in terms of $u$. ‖ ***A*** ∻Key Concept∻ ⚹$u$-substitution⚹ ∻Explanation∻ ⚹In $u$-substitution, we look for a function within the integral whose derivative is also present. This allows us to simplify the integral by substituting $du$ for the corresponding terms in $dx$.⚹◊What method can Heba use to simplify the integral $\int(-12x - 1) \sqrt{-6x^2 - x + 1}dx$ through $u$-substitution?⍭ Generate me a similar question◊

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